Answer
$$y''=2\csc^2(3x-1)\cot(3x-1)$$
Work Step by Step
$$y=\frac{1}{9}\cot(3x-1)$$
- Find $y'$: $$y'=\frac{1}{9}\Big(\cot(3x-1)\Big)'=\frac{1}{9}\Big(-\csc^2(3x-1)\Big)(3x-1)'$$ $$y'=-\frac{1}{9}\csc^2(3x-1)\times3=-\frac{1}{3}\csc^2(3x-1)$$
- Find $y''$: $$y''=-\frac{1}{3}\Big(\csc^2(3x-1)\Big)'=-\frac{1}{3}\times2\csc(3x-1)\Big(\csc(3x-1)\Big)'$$ $$y''=-\frac{2}{3}\csc(3x-1)\Big(-\csc(3x-1)\cot(3x-1)(3x-1)'\Big)$$ $$y''=\frac{2}{3}\csc^2(3x-1)\cot(3x-1)\times3$$ $$y''=2\csc^2(3x-1)\cot(3x-1)$$
